Noncritical Nonrelativistic String Theory

• Noncritical nonrelativistic string theory is a formulation that extends traditional string models by incorporating extra worldsheet degrees of freedom and a composite linear dilaton to achieve conformal invariance.
• The methodology employs modified worldsheet dynamics with additional ghost fields and a composite linear dilaton term to cancel anomalies without requiring a critical target-space dimension.
• The implications include establishing dualities with two-dimensional Yang-Mills theory and enabling consistent off-critical string backgrounds through generalized spacetime equations.

A noncritical version of nonrelativistic string theory refers to the extension of nonrelativistic string models beyond the traditional target-space critical dimension by incorporating extra worldsheet degrees of freedom that restore conformal invariance. Such constructions generalize the original framework by Gomis and Ooguri, which demanded criticality, and enable the formulation of consistent string backgrounds in arbitrary (typically lower) spacetime dimensions, often realized via the introduction of a linear or composite linear dilaton. This approach has been applied to provide string duals of large-$N_c$ chiral two-dimensional Yang-Mills theory and to establish spacetime equations for nonrelativistic string theory in background fields (Komatsu et al., 30 Nov 2025, Gomis et al., 2019).

1. Worldsheet Structure and the Composite Linear Dilaton

In the Komatsu–Maity noncritical nonrelativistic string model, the worldsheet theory is constructed in conformal gauge on the sphere, utilizing complex coordinates $(z, \bar z)$. The fundamental matter fields are $X^\pm(z, \bar z)$ and their conjugate $\beta$-$\gamma$ partners, $\beta(z), \bar\beta(\bar z)$, together with holomorphic $(b,c)$ ghosts. The action reads

$$S = \int \frac{d^2z}{2\pi} \left[ \beta\,\bar\partial X^{+} + \bar\beta\,\partial X^{-} + \frac{q}{2} \mathcal{L}_{\rm CLD} + \frac{\lambda\pi}{2}\left(\partial X^{+}\,\bar\partial X^{-} - \bar\partial X^{+}\,\partial X^{-}\right)\right] + S_{bc}$$

where the “tensionful” term, proportional to $\lambda$, is an area form that replicates the $B$-field-like coupling in the Gomis–Ooguri nonrelativistic string but is topological in nature and does not contribute to the stress tensor.

The composite linear dilaton (CLD) Lagrangian is a distinctive feature: $$\mathcal{L}_{\rm CLD} = 2\,\partial\varphi\,\bar\partial\varphi + \hat R\,\varphi, \qquad \varphi = \log[\partial X^+\,\bar\partial X^- R^2]$$ Here, $\hat R$ denotes the worldsheet Ricci scalar, and $R$ is the target-space circle radius. The CLD structure replaces the usual free Liouville field with a dynamical variable dependent on the worldsheet derivatives of $X^+$ and $X^-$, rendering the model highly nontrivial and directly tied to the covering-map data of the dual gauge theory (Komatsu et al., 30 Nov 2025).

2. Central Charge, Anomaly Cancellation, and Weyl Invariance

The conformal structure is governed by the interplay between the matter content and the ghost system. Specifically, the kinetic $\beta$-$\gamma$ sector contributes a central charge $c=2$, while the CLD term, due to its effective background charge $Q\sim\sqrt{2q}$, adds $12q$ to the total. The matter central charge is

$c_{\rm matter} = 2 + 12q$

For conformal invariance, the combined matter and ghost anomaly must cancel, leading to

$$c_{\rm matter} + c_{bc} = 0 \implies 2 + 12q - 26 = 0 \implies q = 1$$

Thus, the theory achieves worldsheet conformal invariance without the critical-dimension constraint by virtue of the nontrivial CLD background, demonstrating the noncritical realization of a nonrelativistic string (Komatsu et al., 30 Nov 2025).

In the broader nonrelativistic sigma model context, as treated in (Gomis et al., 2019), the total Weyl anomaly cancellation involves the introduction of a Liouville (or linear-dilaton) mode when the target space dimension is less than 26, enabling the construction of a consistent noncritical theory with generalized spacetime equations encompassing the Liouville field as an extra target-space direction.

3. Worldsheet Operator Product Expansions and Stress Tensor

The fundamental OPEs in the noncritical nonrelativistic setup are

$$\beta(z)\,X^+(w) \sim -\frac{1}{z-w}, \qquad \bar\beta(\bar z)\,X^-(\bar w) \sim -\frac{1}{\bar z - \bar w}$$

However, the presence of the composite linear dilaton term modifies the OPE structure, in particular generating a nonstandard $\beta \times \beta$ OPE: $$\beta(z)\,\beta(w) \sim 2q\,\partial_z \partial_w\left[ \frac{1}{(z-w)^2} \frac{1}{\partial_z X^+(z) \partial_w X^+(w)} \right]$$ The holomorphic stress tensor realizes the quantum Virasoro algebra with the shifted central charge: $T(z)= -\beta\,\partial X^+ + 2q\,\{X^+,z\}$ with $$\{X^+,z\} = \partial^2 \log \partial X^+ - \frac{1}{2} (\partial \log \partial X^+)^2$$ and

$T(z)T(w) \sim \frac{c/2}{(z-w)^4} + \cdots$

confirming the closure of worldsheet conformal symmetry for the noncritical theory (Komatsu et al., 30 Nov 2025).

4. Vertex Operator Construction and Physical Spectrum

The only local physical degrees of freedom on the worldsheet are winding modes around the compact target-space direction associated with $X^1$. The canonical winding-$w_k$ vertex operator of energy $E_k = \pi\lambda R\,w_k$ is

$$\mathcal{V}_k(z_k,\bar z_k) = \exp\left[ i\,w_k R \int^{z_k} (\beta\,dz - \bar\beta\,d\bar z) - i\,E_k X^0(z_k, \bar z_k) \right]$$

These operators are primaries of conformal weight $(1,1)$ when the energy satisfies the on-shell constraint, and BRST invariance coincides with this primary condition together with the winding-momentum relation. This structure parallels the physical state conditions in conventional string theory but is adapted to the nonrelativistic, noncritical context (Komatsu et al., 30 Nov 2025).

5. Scattering Amplitudes and Localization Phenomena

Worldsheet correlation functions with such vertex insertions localize, via path integration, onto classical solutions characterized by so-called 'Mandelstam maps': $$X^+(z) \to R \rho(z),~X^-(\bar z) \to R\bar\rho(\bar z),~\rho(z) = -i \sum_{k=1}^n w_k \log(z-z_k)$$ The resulting n-point amplitude includes a nontrivial determinant factor (from the $\beta\bar\beta$ Gaussian integration) and the CLD action evaluated on the classical solution: $$\left\langle\prod_{k=1}^n\mathcal{V}_k\right\rangle = g_s^{n-2} (4\pi^2 R)\,\delta\big(\sum E_k\big)\,\delta_{\sum w_k} \prod_{k=1}^n (w_k)^{q} \prod_{i<j} |z_i - z_j|^{-2q} |\Delta(P_n)|^q$$ with $P_n(z) = \sum_k w_k \prod_{i\neq k} (z-z_i)$, and the three- and four-point amplitudes explicitly reproduce the structure of chiral large-$N$ 2d Yang-Mills theory at finite coupling, validating the duality proposal (Komatsu et al., 30 Nov 2025).

6. Comparison with the Critical Gomis–Ooguri Model

The original Gomis–Ooguri nonrelativistic string theory is characterized by a $\beta$-$\gamma$ worldsheet system and a tensionful term $\partial X^+\wedge\partial X^-$, but is restricted to critical target-space dimension ($d=26$ for the bosonic sector), owing to the need for vanishing total Weyl anomaly. The key advance in the noncritical version is the introduction of the composite linear dilaton, which provides an additional central charge necessary for anomaly cancellation in lower dimensions. This enables one to construct fully consistent, off-critical, nonrelativistic worldsheet CFTs (Komatsu et al., 30 Nov 2025).

The general noncritical formulation in Newton–Cartan backgrounds, as analyzed in (Gomis et al., 2019), employs a Liouville/linaer-dilaton compensator to absorb the central charge deficit for $d<26$. In both approaches, the Liouville or CLD term plays the central role of balancing the central charge and restoring conformal invariance, with the spacetime background equations universally modified to incorporate the new dilaton gradient.

7. Generalized Spacetime Equations for Noncritical Nonrelativistic Strings

The presence of the Liouville or CLD mode alters the spacetime equations derived from worldsheet Weyl invariance. For a general nonrelativistic sigma model with Newton–Cartan data $\tau_\mu{}^A, H_{\mu\nu}, B_{\mu\nu}, \Phi$ and introducing $\Phi_{\rm tot}(x,\varphi) = \Phi(x) + V\varphi$, the generalized field equations are

$$\begin{aligned} &D_{[\mu}\tau_{\nu]}{}^A=0 \ &R_{\mu\nu} + 2\nabla_\mu\nabla_\nu\Phi_{\rm tot} - \tfrac14 \mathcal{H}_{\mu\rho\sigma}\mathcal{H}_\nu{}^{\rho\sigma} = 0 \ &\nabla^\rho \mathcal{H}_{\rho\mu\nu} - 2\nabla^\rho\Phi_{\rm tot} \mathcal{H}_{\rho\mu\nu} = 0 \ &\nabla^2\Phi_{\rm tot} - (\nabla\Phi_{\rm tot})^2 + \tfrac14 R - \tfrac1{48} \mathcal{H}^2 = 0 \end{aligned}$$

These field equations characterize a one-parameter family (slope $V$) of Weyl-invariant noncritical nonrelativistic string backgrounds and generalize the conventional critical string Newton–Cartan systems (Gomis et al., 2019). The effective dilaton’s gradient introduces new geometric effects while preserving worldsheet consistency.

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References:

• "Chiral Composite Linear Dilaton as String Dual to Two-Dimensional Yang-Mills" (Komatsu et al., 30 Nov 2025)
• "Nonrelativistic String Theory in Background Fields" (Gomis et al., 2019)